1. Introduction

To achieve the thermonuclear condition (the ignition condition) in a tokamaks [1-10] , it is necessary to confine the plasma for a sufficient long time. The global energy confinement time , is defined by , where n is the plasma density, T_e and T_i are the electron and ion plasma temperature and P is the total input power [11-12].

The main purpose of our research work is to understand the theory of confinement in tokomaks. It is necessary to introduce different empirical methods and combine data from different tokamaks which operates in different conditions and by using statistical methods to show the dependence of the confinement time on the different plasma parameters which provides scaling relations for different tokamaks [3-10].

In this paper we present the dependence of effective edge safety factor on the energy confinement time by analytical solution of Grad-Shafranov equation (GSE) [1-2] with Lithium limiter [13] for circular cross-section HT-7 tokamak [8-9]. For this, the Shafranov parameter was obtained from analytical solution of GSE [4, 7]. It is observed, the energy maximum energy confinement time relate to the low values of effective edge safety factor .

2. Theoretical Equation

For circular cross-section HT-7 tokamak [7-9, 13], which is the ohmically heated tokamak, the Grad–Shafranov equation is solved by formally expanding as follows [5, 7, 12]:

(1)

(2)

(3)

where is major radius, is the vacuum toroidal field at is a new free function replacing . In the first order solution or toroidal force balance approximation and if plasma were surrounded by a perfectly conducting shell located at , then first order flux function [5, 7, 12] is:

(4)

If there are external coils to produce vertical magnetic field, the boundary condition on the flux function is modified so that we have [12]:

(5)

where is the flux function due to external vertical field coils and therefore the full toroidal correction to is [5, 7, 12]:

(6)

The shift of the plasma column center from the geometrical center of vacuum chamber given by [5, 7, 12]:

(7)

where .

Therefore, the first relation for plasma position [5, 7, 12] is

(8)

where is the poloidal beta, is the internal inductance of the plasma, and is the average vertical magnetic field over the vacuum chamber. We can find from saddle sine coil [9] and expression from magnetic coils measurement [9]:

(9)

where

(10)

We measured these local magnetic fields with magnetic probes [9] at above angles.

Therefore the effective edge safety factor at the plasma edge is given by [14]:

(11)

From this relation, using the value of Shafranov parameter obtained from analytical solution of GSE [7], we plotted time history of the effective edge safety factor in target shot for circular cross section HT-7 tokamak, as shown in Figure 1.

3. Experimental Work

Before we derive the relation for the energy confinement time, we must determine the volume-averaged plasma kinetic pressure , and then the plasma thermal energy U. can be determined directly from the definition of the poloidal beta [4]:

(12)

where a is the plasma minor radius. For the measurement of the plasma thermal energy, we start from the plasma state equation [4]:

(13)

where subscript ‘i’ indicates the plasma species i and E indicates the plasma thermal energy density; therefore the plasma thermal energy U and the plasma temperature are obtained [4]:

(14)

where V is the plasma volume. The plasma-specific resistance in the steady state plasma can be written as [4]

(15)

where is the plasma conductivity, is the plasma resistance and is the resistive component of the loop voltage (the poloidal flux loop).

But the most important of these measurements is determining the plasma thermal energy confinement time, which is defined by [4]

(16)

where is the plasma energy confinement time and is the rate of input heating power. Rearranging equation (16), the ohmic heating power is [4]

(17)

If the plasma is in thermal equilibrium ( and ), then from equations (14) and (16), we have

(18)

(19)

Also if is not negligible, then from equations (14) and (16), we have

Figure 1. The dependence of effective edge safety factor on energy confinement time, obtained analytically by solution of GSE

(20)

Therefore, according to the above discussion, the Shafranov parameter was obtained from by analytical solution of GSE. It is observed, the maximum energy confinement time relate to the low values of effective edge safety factor .

4. Result and Discussion

The dependence of effective edge safety factor on the energy confinement time have been studied by analytical solution of Grad-Shafranov equation (GSE) with Lithium limiter [13], for circular cross-section HT-7 tokamak. It is observed, the maximum energy confinement time relate to the low values of effective edge safety factor .